\section{Noter fra MM4}

\subsection*{Opgave 1}

Which of the propositions are correct (why):

a) If cakes lack yeast, then they become small.
The cakes are small. So they lack yeast.
b) If cakes lack yeast, then they become small.
The cakes do not lack yeast. So they do not become small.
c) If cakes lack yeast, then they become small.
The cakes do not become small. So they do not lack yeast.

Svar:

A og B er forkerte da de antager at manglende gær er den eneste årsag til at kagerne bliver små. 
C er absolut rigtigt.


Induction or deduction?

1. All throughout history people repeat the same mistakes, so we can conclude that mistakes will be made in the future.

A: Induktion

2. The whale is a mammal, so all killer whales are mammals.

A: Deduktion, under antagelsen af at killer whales er whales.

3. All killer whales are mammals, so the whale is a mammal.

A: Induktion, fordi ikke alle whales er tjekket for at være mammals.

4. Dumbbell training is inherently safe.  I've never observed a torn muscle or any other serious injury resulting from the proper use of dumbbells. (Bill Philips and Michael D'Orso, Body for Life (New York: Harper Collins, 1999), 123.)

A: Induktion, gjort ud fra observationer.

5. All human beings have the ability to think rationally and realistically. We all can realize, "Even if I am probably correct, there is still room for questioning." Thus we can allow discussion, disconfirmation, and new evidence to change our minds. (James O. Prochaska, et at., Changing for Good (New York: William Morrrow, 1994), 182.)

A: Deduktion

6. Most psychohistorians reject non-psychoanalytic psychologies for use in historical research because of their ahistorical non-developmental character and because they are either so simplistic that they explain only elementary traits or so locking in structural coherence as to be unusable by historians. (Peter Loewenberg, Decoding the Past: the Psychohistorical Approach (New York: Alfred A. Knopf, 1983), 19.)

A: Induktion

7. In logic we are concerned with propositions rather than beliefs, since logic is not interested in what people do in fact believe, only in the conditions which determine the truth or falsehood of possible beliefs. (Bertrand Russell, The Analysis of Mind (London: Routledge, 1989), 204.)

Deduktiv

8. Many of those children whose conduct have been most narrowly watched, become the weakest men, because their instructors only instill certain notions into their minds, that have no other foundation than their authority and if they be loved or respected, the mind is cramped in its exertions and wavering in its advances. (Mary Wollstonecraft, Vindication of the Rights of Woman (1792 London: T. Fisher Unwin, 1891), 45.)

Induktiv

9. Given the view that species evolve into one another, then members of one species must somehow give rise to members of another species.  It follows that members of the second species must somehow derive as variants of members of the first. (Stuart A. Kauffman, The Origins of Order: Self-Organization and Selection in Evolution (Oxford: Oxford University Press, 1993), 6.)

Deduktiv

10. Astronomers infer the existence of dark matter because it would provide the unseen glue that keeps galaxies intact and galaxy clusters from disassembling. But dark matter has never been convincingly detected directly. (Ron Cowen, $"$PAMELA Spots the Dark Stuff, Maybe,$"$ Science News (2008)  vol. 174:3, 8.)

Induktiv

\subsection*{Opgave 2}

Tautology: If for any instatiation L,A,C with T and F -A is true.

Prove the following proposition using the truth table:

a) (p → (q→p)) is a tautology

\begin{tabular}{|c|c|c|c|}
\hline q & p & q$\rightarrow$ p & p$\rightarrow$ (q$\rightarrow$ p) \\ 
\hline T & T & T & T \\ 
\hline T & F & T & T \\ 
\hline F & T & F & T \\
\hline F & F & T & T \\
\hline 
\end{tabular} 

b) (p→(q→r)) → ((p→q)→(p→r)) is a tautology

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline p & q & r & q$\rightarrow$r & p$\rightarrow$q & p$\rightarrow$r & p$\rightarrow$(q$\rightarrow$r) & (p$\rightarrow$q)$\rightarrow$(p$\rightarrow$r) & (p$\rightarrow$(q$\rightarrow$r))$\rightarrow$((p$\rightarrow$q)$\rightarrow$(p$\rightarrow$r)) \\ 
\hline T & T & T & T & T & T & T & T & T \\ 
\hline T & T & F & F & T & F & F & F & T \\ 
\hline T & F & T & T & F & T & T & T & T \\ 
\hline F & F & F & T & T & T & T & T & T \\ 
\hline F & T & T & T & T & T & T & T & T \\ 
\hline F & T & F & F & T & T & T & T & T \\ 
\hline 
\end{tabular} 